Properties

Label 1936.4.a.l
Level $1936$
Weight $4$
Character orbit 1936.a
Self dual yes
Analytic conductor $114.228$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1936 = 2^{4} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1936.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(114.227697771\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{3} - 2q^{5} + 24q^{7} - 11q^{9} + O(q^{10}) \) \( q + 4q^{3} - 2q^{5} + 24q^{7} - 11q^{9} - 22q^{13} - 8q^{15} - 50q^{17} + 44q^{19} + 96q^{21} + 56q^{23} - 121q^{25} - 152q^{27} - 198q^{29} + 160q^{31} - 48q^{35} - 162q^{37} - 88q^{39} + 198q^{41} + 52q^{43} + 22q^{45} - 528q^{47} + 233q^{49} - 200q^{51} - 242q^{53} + 176q^{57} + 668q^{59} - 550q^{61} - 264q^{63} + 44q^{65} - 188q^{67} + 224q^{69} - 728q^{71} - 154q^{73} - 484q^{75} - 656q^{79} - 311q^{81} + 236q^{83} + 100q^{85} - 792q^{87} + 714q^{89} - 528q^{91} + 640q^{93} - 88q^{95} - 478q^{97} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 4.00000 0 −2.00000 0 24.0000 0 −11.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1936.4.a.l 1
4.b odd 2 1 968.4.a.a 1
11.b odd 2 1 16.4.a.a 1
33.d even 2 1 144.4.a.e 1
44.c even 2 1 8.4.a.a 1
55.d odd 2 1 400.4.a.g 1
55.e even 4 2 400.4.c.i 2
77.b even 2 1 784.4.a.e 1
88.b odd 2 1 64.4.a.b 1
88.g even 2 1 64.4.a.d 1
132.d odd 2 1 72.4.a.c 1
176.i even 4 2 256.4.b.a 2
176.l odd 4 2 256.4.b.g 2
220.g even 2 1 200.4.a.g 1
220.i odd 4 2 200.4.c.e 2
264.m even 2 1 576.4.a.j 1
264.p odd 2 1 576.4.a.k 1
308.g odd 2 1 392.4.a.e 1
308.m odd 6 2 392.4.i.b 2
308.n even 6 2 392.4.i.g 2
396.k even 6 2 648.4.i.h 2
396.o odd 6 2 648.4.i.e 2
440.c even 2 1 1600.4.a.o 1
440.o odd 2 1 1600.4.a.bm 1
572.b even 2 1 1352.4.a.a 1
660.g odd 2 1 1800.4.a.d 1
660.q even 4 2 1800.4.f.u 2
748.f even 2 1 2312.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.a.a 1 44.c even 2 1
16.4.a.a 1 11.b odd 2 1
64.4.a.b 1 88.b odd 2 1
64.4.a.d 1 88.g even 2 1
72.4.a.c 1 132.d odd 2 1
144.4.a.e 1 33.d even 2 1
200.4.a.g 1 220.g even 2 1
200.4.c.e 2 220.i odd 4 2
256.4.b.a 2 176.i even 4 2
256.4.b.g 2 176.l odd 4 2
392.4.a.e 1 308.g odd 2 1
392.4.i.b 2 308.m odd 6 2
392.4.i.g 2 308.n even 6 2
400.4.a.g 1 55.d odd 2 1
400.4.c.i 2 55.e even 4 2
576.4.a.j 1 264.m even 2 1
576.4.a.k 1 264.p odd 2 1
648.4.i.e 2 396.o odd 6 2
648.4.i.h 2 396.k even 6 2
784.4.a.e 1 77.b even 2 1
968.4.a.a 1 4.b odd 2 1
1352.4.a.a 1 572.b even 2 1
1600.4.a.o 1 440.c even 2 1
1600.4.a.bm 1 440.o odd 2 1
1800.4.a.d 1 660.g odd 2 1
1800.4.f.u 2 660.q even 4 2
1936.4.a.l 1 1.a even 1 1 trivial
2312.4.a.a 1 748.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1936))\):

\( T_{3} - 4 \)
\( T_{5} + 2 \)
\( T_{7} - 24 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 4 T + 27 T^{2} \)
$5$ \( 1 + 2 T + 125 T^{2} \)
$7$ \( 1 - 24 T + 343 T^{2} \)
$11$ 1
$13$ \( 1 + 22 T + 2197 T^{2} \)
$17$ \( 1 + 50 T + 4913 T^{2} \)
$19$ \( 1 - 44 T + 6859 T^{2} \)
$23$ \( 1 - 56 T + 12167 T^{2} \)
$29$ \( 1 + 198 T + 24389 T^{2} \)
$31$ \( 1 - 160 T + 29791 T^{2} \)
$37$ \( 1 + 162 T + 50653 T^{2} \)
$41$ \( 1 - 198 T + 68921 T^{2} \)
$43$ \( 1 - 52 T + 79507 T^{2} \)
$47$ \( 1 + 528 T + 103823 T^{2} \)
$53$ \( 1 + 242 T + 148877 T^{2} \)
$59$ \( 1 - 668 T + 205379 T^{2} \)
$61$ \( 1 + 550 T + 226981 T^{2} \)
$67$ \( 1 + 188 T + 300763 T^{2} \)
$71$ \( 1 + 728 T + 357911 T^{2} \)
$73$ \( 1 + 154 T + 389017 T^{2} \)
$79$ \( 1 + 656 T + 493039 T^{2} \)
$83$ \( 1 - 236 T + 571787 T^{2} \)
$89$ \( 1 - 714 T + 704969 T^{2} \)
$97$ \( 1 + 478 T + 912673 T^{2} \)
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